Optimal. Leaf size=29 \[ \frac {\log \left (\frac {x}{2}\right ) \left (\log (x)+\frac {e^x}{2 x-\log (\log (\log (4)))}\right )}{x} \]
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Rubi [B] time = 3.88, antiderivative size = 62, normalized size of antiderivative = 2.14, number of steps used = 26, number of rules used = 11, integrand size = 155, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 27, 6688, 6742, 14, 2304, 2303, 2366, 2178, 2177, 2554} \begin {gather*} \frac {\log (x) \log \left (\frac {x}{2}\right )}{x}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{\log (\log (\log (4))) (2 x-\log (\log (\log (4))))} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 27
Rule 1594
Rule 2177
Rule 2178
Rule 2303
Rule 2304
Rule 2366
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{x^2 \left (4 x^2-4 x \log (\log (\log (4)))+\log ^2(\log (\log (4)))\right )} \, dx\\ &=\int \frac {2 e^x x+\left (4 x^2+e^x \left (-4 x+2 x^2\right )\right ) \log \left (\frac {x}{2}\right )+\left (4 x^2-4 x^2 \log \left (\frac {x}{2}\right )\right ) \log (x)+\left (-e^x+\left (e^x (1-x)-4 x\right ) \log \left (\frac {x}{2}\right )+\left (-4 x+4 x \log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log (\log (\log (4)))+\left (\log \left (\frac {x}{2}\right )+\left (1-\log \left (\frac {x}{2}\right )\right ) \log (x)\right ) \log ^2(\log (\log (4)))}{x^2 (2 x-\log (\log (\log (4))))^2} \, dx\\ &=\int \frac {\left (e^x+\log (x) (2 x-\log (\log (\log (4))))\right ) (2 x-\log (\log (\log (4))))+\log \left (\frac {x}{2}\right ) \left ((-2 x+\log (\log (\log (4))))^2-\log (x) (-2 x+\log (\log (\log (4))))^2+e^x \left (2 x^2+\log (\log (\log (4)))-x (4+\log (\log (\log (4))))\right )\right )}{x^2 (2 x-\log (\log (\log (4))))^2} \, dx\\ &=\int \left (\frac {\log \left (\frac {x}{2}\right )+\log (x)-\log \left (\frac {x}{2}\right ) \log (x)}{x^2}+\frac {e^x \left (2 x+2 x^2 \log \left (\frac {x}{2}\right )-4 x \log \left (\frac {x}{2}\right ) \left (1+\frac {1}{4} \log (\log (\log (4)))\right )-\log (\log (\log (4)))+\log \left (\frac {x}{2}\right ) \log (\log (\log (4)))\right )}{x^2 (2 x-\log (\log (\log (4))))^2}\right ) \, dx\\ &=\int \frac {\log \left (\frac {x}{2}\right )+\log (x)-\log \left (\frac {x}{2}\right ) \log (x)}{x^2} \, dx+\int \frac {e^x \left (2 x+2 x^2 \log \left (\frac {x}{2}\right )-4 x \log \left (\frac {x}{2}\right ) \left (1+\frac {1}{4} \log (\log (\log (4)))\right )-\log (\log (\log (4)))+\log \left (\frac {x}{2}\right ) \log (\log (\log (4)))\right )}{x^2 (2 x-\log (\log (\log (4))))^2} \, dx\\ &=\int \left (\frac {\log \left (\frac {x}{2}\right )}{x^2}-\frac {\left (-1+\log \left (\frac {x}{2}\right )\right ) \log (x)}{x^2}\right ) \, dx+\int \frac {e^x \left (2 x-\log (\log (\log (4)))+\log \left (\frac {x}{2}\right ) \left (2 x^2+\log (\log (\log (4)))-x (4+\log (\log (\log (4))))\right )\right )}{x^2 (2 x-\log (\log (\log (4))))^2} \, dx\\ &=\int \frac {\log \left (\frac {x}{2}\right )}{x^2} \, dx-\int \frac {\left (-1+\log \left (\frac {x}{2}\right )\right ) \log (x)}{x^2} \, dx+\int \left (\frac {e^x}{x^2 (2 x-\log (\log (\log (4))))}+\frac {e^x \log \left (\frac {x}{2}\right ) \left (2 x^2+\log (\log (\log (4)))-x (4+\log (\log (\log (4))))\right )}{x^2 (2 x-\log (\log (\log (4))))^2}\right ) \, dx\\ &=-\frac {1}{x}-\frac {\log \left (\frac {x}{2}\right )}{x}+\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\int \frac {\log \left (\frac {x}{2}\right )}{x^2} \, dx+\int \frac {e^x}{x^2 (2 x-\log (\log (\log (4))))} \, dx+\int \frac {e^x \log \left (\frac {x}{2}\right ) \left (2 x^2+\log (\log (\log (4)))-x (4+\log (\log (\log (4))))\right )}{x^2 (2 x-\log (\log (\log (4))))^2} \, dx\\ &=\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{(2 x-\log (\log (\log (4)))) \log (\log (\log (4)))}+\int \left (-\frac {2 e^x}{x \log ^2(\log (\log (4)))}+\frac {4 e^x}{(2 x-\log (\log (\log (4)))) \log ^2(\log (\log (4)))}-\frac {e^x}{x^2 \log (\log (\log (4)))}\right ) \, dx-\int \frac {e^x}{x^2 (2 x-\log (\log (\log (4))))} \, dx\\ &=\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{(2 x-\log (\log (\log (4)))) \log (\log (\log (4)))}-\frac {2 \int \frac {e^x}{x} \, dx}{\log ^2(\log (\log (4)))}+\frac {4 \int \frac {e^x}{2 x-\log (\log (\log (4)))} \, dx}{\log ^2(\log (\log (4)))}-\frac {\int \frac {e^x}{x^2} \, dx}{\log (\log (\log (4)))}-\int \left (-\frac {2 e^x}{x \log ^2(\log (\log (4)))}+\frac {4 e^x}{(2 x-\log (\log (\log (4)))) \log ^2(\log (\log (4)))}-\frac {e^x}{x^2 \log (\log (\log (4)))}\right ) \, dx\\ &=\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\frac {2 \text {Ei}(x)}{\log ^2(\log (\log (4)))}+\frac {2 \text {Ei}\left (\frac {1}{2} (2 x-\log (\log (\log (4))))\right ) \sqrt {\log (\log (4))}}{\log ^2(\log (\log (4)))}+\frac {e^x}{x \log (\log (\log (4)))}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{(2 x-\log (\log (\log (4)))) \log (\log (\log (4)))}+\frac {2 \int \frac {e^x}{x} \, dx}{\log ^2(\log (\log (4)))}-\frac {4 \int \frac {e^x}{2 x-\log (\log (\log (4)))} \, dx}{\log ^2(\log (\log (4)))}+\frac {\int \frac {e^x}{x^2} \, dx}{\log (\log (\log (4)))}-\frac {\int \frac {e^x}{x} \, dx}{\log (\log (\log (4)))}\\ &=\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\frac {\text {Ei}(x)}{\log (\log (\log (4)))}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{(2 x-\log (\log (\log (4)))) \log (\log (\log (4)))}+\frac {\int \frac {e^x}{x} \, dx}{\log (\log (\log (4)))}\\ &=\frac {\log \left (\frac {x}{2}\right ) \log (x)}{x}-\frac {e^x \log \left (\frac {x}{2}\right )}{x \log (\log (\log (4)))}+\frac {2 e^x \log \left (\frac {x}{2}\right )}{(2 x-\log (\log (\log (4)))) \log (\log (\log (4)))}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 29, normalized size = 1.00 \begin {gather*} \frac {\log \left (\frac {x}{2}\right ) \left (\log (x)+\frac {e^x}{2 x-\log (\log (\log (4)))}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 63, normalized size = 2.17 \begin {gather*} \frac {2 \, x \log \left (\frac {1}{2} \, x\right )^{2} + {\left (2 \, x \log \relax (2) + e^{x}\right )} \log \left (\frac {1}{2} \, x\right ) - {\left (\log \relax (2) \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2}\right )} \log \left (\log \left (2 \, \log \relax (2)\right )\right )}{2 \, x^{2} - x \log \left (\log \left (2 \, \log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 71, normalized size = 2.45 \begin {gather*} -\frac {2 \, x \log \relax (2) \log \relax (x) - 2 \, x \log \relax (x)^{2} - \log \relax (2) \log \relax (x) \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right ) + \log \relax (x)^{2} \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right ) + e^{x} \log \relax (2) - e^{x} \log \relax (x)}{2 \, x^{2} - x \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 47, normalized size = 1.62
method | result | size |
default | \(\frac {{\mathrm e}^{x} \ln \relax (2)-{\mathrm e}^{x} \ln \relax (x )}{x \left (\ln \left (\ln \left (2 \ln \relax (2)\right )\right )-2 x \right )}-\frac {\ln \relax (2) \ln \relax (x )}{x}+\frac {\ln \relax (x )^{2}}{x}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 70, normalized size = 2.41 \begin {gather*} \frac {{\left (2 \, x - \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )\right )} \log \relax (x)^{2} - {\left (\log \relax (2) - \log \relax (x)\right )} e^{x} - {\left (2 \, x \log \relax (2) - \log \relax (2) \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )\right )} \log \relax (x)}{2 \, x^{2} - x \log \left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \relax (x)\,\left (4\,x^2\,\ln \left (\frac {x}{2}\right )-4\,x^2\right )-2\,x\,{\mathrm {e}}^x+\ln \left (\ln \left (2\,\ln \relax (2)\right )\right )\,\left ({\mathrm {e}}^x+\ln \relax (x)\,\left (4\,x-4\,x\,\ln \left (\frac {x}{2}\right )\right )+\ln \left (\frac {x}{2}\right )\,\left (4\,x+{\mathrm {e}}^x\,\left (x-1\right )\right )\right )+\ln \left (\frac {x}{2}\right )\,\left ({\mathrm {e}}^x\,\left (4\,x-2\,x^2\right )-4\,x^2\right )-{\ln \left (\ln \left (2\,\ln \relax (2)\right )\right )}^2\,\left (\ln \left (\frac {x}{2}\right )-\ln \relax (x)\,\left (\ln \left (\frac {x}{2}\right )-1\right )\right )}{4\,x^4-4\,\ln \left (\ln \left (2\,\ln \relax (2)\right )\right )\,x^3+{\ln \left (\ln \left (2\,\ln \relax (2)\right )\right )}^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 41, normalized size = 1.41 \begin {gather*} \frac {\left (\log {\relax (x )} - \log {\relax (2 )}\right ) e^{x}}{2 x^{2} - x \log {\left (\log {\left (\log {\relax (2 )} \right )} + \log {\relax (2 )} \right )}} + \frac {\log {\relax (x )}^{2}}{x} - \frac {\log {\relax (2 )} \log {\relax (x )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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